Once it is found the stress distribution, it is important to understand the correlation between load and stress. Actually, loads are obtained after the measure of the displacements. Then through a series of equations it is possible to find the membrane loads and moments. This article summarizes the procedure to find the classical lamination theory equation while comments its parameters.

Load – Stress relationship

The loads that can be applied for laminates are characterized by the membrane loads, which are described by N_x, N_y and N_xy. There is no orientation in the z axis, which is the opposite with respect to the one for the beam theory. In general, the axis of the beam, the z axis, is the same as the way of the displacement. This is just a matter of load representation in a positive direction. In addition, there are the moments M_x, M_y and M_xy. The first two are the bending moments, while M_xy is the twisting moment. These are the membrane loads and moments, they are related to the stress inside the laminate.

{N_xy} = {N_x N_y N_xy}^T = ∫_-h/2^h/2{σ_x σ_y σ_xy}^Tdz

{M_xy} = {M_x M_y M_xy}^T = ∫_-h/2^h/2{σ_x σ_y σ_xy}^T∙zdz

Hence, the membrane loads are the stresses integrated across the thickness. In addition, the stresses are given in mega Pascal MPa, Newton per square meter N/m² or Newton per square millimeters, 10⁶ N/mm². Hence, the membrane loads are given by unit length, N/mm. In the case of membrane moments, the unit is mm∙N/mm. Hence, both membrane loads and moments are given per unit length, which is the length of the laminate.

Since the laminate is made by several plies, these are enumerated from the bottommost. The distance from the centerline to a ply is given by h_k and h_k-1, where k is the ply. For instance, the ply 1 is the ply between h1 and h0. In this case, to transform the previous integral to account these plies it is summed the contributions of the n plies.

{N_xy} = Σⁿ₁k{N_xy}_k = Σⁿ₁k ∫^hk_hk-1{σ_xy}_kdz = Σⁿ₁k |Q”|_k∫^hk_hk-1{ε_xy}_kdz

{M_xy} = Σⁿ₁k{M_xy}_k = Σⁿ₁k ∫^hk_hk-1{σ_xy}_kdz = Σⁿ₁k |Q”|_k∫^hk_hk-1{ε_xy}_kdz

Therefore, it is developed equations with the summation of the n-plies of the contribution of each single ply, which is again an integral, but in this case it is not developed along the whole thickness. Actually, it is developed inside the ply, within the limits of the path. Hence, there are some operations, but not with a generic one, instead with the stress within each single ply. This can be related to strain through the stiffness matrix of the ply, which is not the one of the material, because it depends on the orientation. Even though those plies were of the same material, the matrix Q” and K are different for each ply due to the orientation. The reason to relate stress and strain is because this last one is related to the strains of the laminate. The stresses for each ply can be related to the strain. Hence, the stress in a ply is dependent on the strain in the laminate mid plane plus a term, which is given by the curvature times distance of the ply from the mid plane. Since the strain in each ply is known (read more), it can be substituted in the equilibrium equations.

{N_xy} = Σⁿ_{1 k}|Q”|_k⋅{ε_xy}₀⋅∫^hk_hk-1dz + Σⁿ_{1 k}|Q”|_k⋅{K}₀∫^hk_hk-1zdz

{M_xy} = Σⁿ_{1 k}|Q”|_k⋅{ε_xy}₀⋅∫^hk_hk-1dz + Σⁿ_{1 k}|Q”|_k⋅{K}₀∫^hk_hk-1z²dz

The first term is a function of the strains on the mid plane times the integral of the thickness, because this results in the integral of the laminate which is just a summation over the thickness of the laminate. In this case, there is a curvature term of the mid plane times the integral of the distance from the mid plane for each plane, summing overall plies. Hence, for the moment which there are the similar terms:

{N_xy} = Σⁿ_{1 k}|Q”|_k⋅{ε_xy}₀⋅(h_k – h_k-1) + Σⁿ_{1 k}|Q”|_k⋅{K}₀⋅(h_k² – h_k-1²)/2

{M_xy} = Σⁿ_{1 k}|Q”|_k⋅{ε_xy}₀⋅(h_k² – h_k-1²)/2 + Σⁿ_{1 k}|Q”|_k⋅{K}₀⋅(h_k³ – h_k-1³)/3

Finally, this results in a term that instead of the curvature, there are the strains on the mid plane. In addition, there is the curvature that multiplies the integral of the square of the distance across the ply. If these integrals were developed, those equations are reduced to just a summation. Hence, the integrals were completely eliminated from the equilibrium equations. Now it is possible to notice that, there are terms which depending of the summation index k and another one which are not dependent, these are summarised below.

|A| = ∑₁ⁿ _k|Q|_k∙(h_k – h_k-1) ; a_ij = ∑₁ⁿ _k|Q_ij|_k∙(h_k – h_k-1)

|B| = ∑₁ⁿ _k|Q|_k∙(h_k² – h_k-1²)/2 ; b_ij = ∑₁ⁿ k|Q_ij|k∙(h_k² – h_k-1²)/2

|D| = ∑₁ⁿ _k|Q|_k∙(h_k³ – h_k-1³)/3 ; dij = ∑₁ⁿ _k|Q_ij|_k∙(h_k³ – h_k-1³)/3

Where the terms {ε_xy}₀ and {K}₀ are not depending on k, while |A|, |B|, |D|, aij, bij and dij are dependent of k. Since |Q| is a matrix, these are pieces of a longer one. Hence, each of these matrices is composed by elements, in particular, i and j ones, which actually are 3×3 matrices and each cell are described by the formulation above. This depends of the corresponding cell in the three matrices of each ply, thus the single of these matrices is the summation of the corresponding cells in the stiffness matrix k, of n-plies, times the term |Q|.

{NM}^T = {N_x N_y N_xy M_x M_y M_xy}^T ; {ε_xy⁰ k}^T = {ε_x ε_y γ_xy k_x k_y k_xy}^T

{N M}^T = |A B B D|•{ε_xy⁰ k}^T

Finally, putting together the membrane loads and bending moments and the strain and the curvature of the mid plane in a unique vector, the tensorial form is described above. Where the loads {N M} are related to the mid plane strains and curvature {ε_xy⁰ k} and a matrix |A B B D|. This is very compact notation, which is nothing else that a relationship between loads and deformations of a laminate plate. Hence, if there is everything to calculate ABD matrix and there is the information about deformation, it is possible to calculate the load. This is the final form of the classical lamination theory. However, this is not the final result that is necessary, because it is known from loads to deformations. Nevertheless, regarding the stresses on each ply it is required a detailed analysis. This is important to define which is the most critical ply in the laminate. The submatrix A relate the membrane forces with the membrane strains. This is important to know if the deformation on the mid plane is either the extension along y, or the shear deformation, then A would be multiplied by N. D is the relationship between bending/torsion moments and curvatures, this can be obtained by multiplying D by M. The role of B is a coupling between membrane forces and curvature and bending/torsion moments membrane strains. This is an additional term, which if there are moments times B, the membrane strains are obtained, while if there are membrane strains times B, curvatures are obtained. This means that, if the laminate is being bent and B is non-zero, it will be obtained the elongation. In the design point of view, this is not advisable. Another important detail is, to understand the orthotropic equivalent ply for a homogeneous laminate, if it is used the matrix A divided by the thickness of the laminate.

|Q_avg| = |A|/t_lam

This is the average orthotropic stiffness matrix of the laminate. The matrix B is no the best choice, because its coupled behaviour can be confused with respect to the design of the laminate.

Taking the submatrix A for an orthotropic homogeneous material, a cross-ply or balanced lay-up, the formulas obtained are the ones described above. For all of these cases, the formula that represents these conditions is the first one, where X means a populated cell and 0 is a non-populated one. For any other type of lamination, all the cells are populated, which is described by the second formula and will never be zero.

For the matrix D, the formulation is given above. The first matrix represents the orthotropic material, the antisymmetric and cross-ply lay-up configurations. As in the matrix A, the other formulas represent the other lay-ups and it will never be a null matrix. In addition, the second formula can also represent the symmetric lay-up.

For the matrix B, it is non-zero in any case which the lay-up is non-symmetric, because in case of a symmetric lay-up, it is zero, thus no coupling between bending and membrane strains and between membrane loads and curvature. Hence, it could be a good reason to adopt a symmetric lay-up, which the uncoupling condition as suggested. The cross-ply and balanced lay-ups are further additional advantages in eliminating some of the cells. The non-zero cells always are representing something which are related to an internal coupling between the membrane strain and the membrane shear. If there are terms which are non-zero, this means that there is a small internal coupling. Hence, a cross-ply or a balanced behaviour is even better.

References

• P.K. Mallick, Fiber-Reinforced Composites: materials, manufacturing and design – 3° Ed., CRC Press, 2008